Nuprl Lemma : poly-nth-deriv-req
∀[n,d:ℕ]. ∀[a:ℕn + d ⟶ ℝ]. ∀[i:ℕd]. ((poly-nth-deriv(n;a) i) = (r((i + n)!) * (a (i + n)))/(i)!)
Proof
Definitions occuring in Statement :
poly-nth-deriv: poly-nth-deriv(n;a),
int-rdiv: (a)/k1,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
fact: (n)!,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
poly-nth-deriv: poly-nth-deriv(n;a),
less_than': less_than'(a;b),
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermMultiply: left "*" right,
rtermVar: rtermVar(var),
pi1: fst(t),
true: True,
pi2: snd(t),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
poly-deriv: poly-deriv(a),
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
subtract: n - m,
nequal: a ≠ b ∈ T ,
so_apply: x[s],
so_lambda: λ2x.t[x],
int_nzero: ℤ-o,
sq_exists: ∃x:A [B[x]],
rless: x < y,
sq_stable: SqStable(P),
real: ℝ,
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
req_witness,
int_seg_properties,
poly-nth-deriv_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
int-rdiv_wf,
fact_wf,
rmul_wf,
int-to-real_wf,
itermAdd_wf,
int_term_value_add_lemma,
decidable__lt,
int_seg_wf,
real_wf,
istype-nat,
subtract-1-ge-0,
nat_plus_inc_int_nzero,
primrec0_lemma,
add-zero,
rdiv_wf,
int_seg_subtype_nat,
istype-false,
rless-int,
nat_plus_properties,
rless_wf,
assert-rat-term-eq2,
rtermVar_wf,
rtermDivide_wf,
rtermMultiply_wf,
req_functionality,
req_weakening,
int-rdiv-req,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
int_formula_prop_eq_lemma,
intformeq_wf,
satisfiable-full-omega-tt,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
primrec-unroll,
lelt_wf,
add-subtract-cancel,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
add-member-int_seg2,
zero-add,
add-swap,
add-associates,
add-commutes,
int_subtype_base,
le_wf,
false_wf,
nat_plus_wf,
equal-wf-base,
nequal_wf,
less_than_wf,
subtype_rel_sets,
rmul_functionality,
decidable__equal_int,
fact-non-zero,
rneq-int,
equal-wf-T-base,
fact_unroll_1,
req-int,
rmul-int,
int_entire_a,
rinv_wf2,
req_wf,
rneq_functionality,
sq_stable__less_than,
rdiv_functionality,
uiff_transitivity,
rinv-of-rmul,
req_transitivity,
real_term_polynomial,
itermMultiply_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rmul-rinv3,
rinv-mul-as-rdiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
productElimination,
imageElimination,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
applyEquality,
dependent_set_memberEquality_alt,
unionElimination,
because_Cache,
addEquality,
productIsType,
functionIsType,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
inrFormation_alt,
applyLambdaEquality,
cumulativity,
instantiate,
promote_hyp,
computeAll,
intEquality,
lambdaEquality,
dependent_pairFormation,
equalityElimination,
lambdaFormation,
voidEquality,
isect_memberEquality,
dependent_set_memberEquality,
baseClosed,
setEquality,
functionExtensionality,
inrFormation,
multiplyEquality,
closedConclusion,
baseApply,
imageMemberEquality
Latex:
\mforall{}[n,d:\mBbbN{}]. \mforall{}[a:\mBbbN{}n + d {}\mrightarrow{} \mBbbR{}]. \mforall{}[i:\mBbbN{}d]. ((poly-nth-deriv(n;a) i) = (r((i + n)!) * (a (i + n)))/(i)!)
Date html generated:
2019_10_30-AM-09_02_16
Last ObjectModification:
2019_04_02-AM-09_46_45
Theory : reals
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